If f (x) is continuous and f((9)/(2)) = (2)/(9), then _ x → 0 f( 1...

Given: $f\left(\dfrac{9}{2}\right) = \dfrac{2}{9} and$f\left(x\right)$is continuous.<br /><br />$\lim_{x \to 0}f\left(\dfrac{1 - \cos 3x}{{x}^{2}}\right) = f\left(\lim_{x \to 0}\dfrac{1 - \cos 3x}{{x}^{2}}\right)$<br /><br />$= f\left(\lim_{x \to 0}\dfrac{2\sin^{2}\left(\dfrac{3x}{2}\right)}{{x}^{2}}\right)$\left\{∵ 1 - \cos 2\theta = 2\sin^{2}\theta\right\}$

$= f\left(\lim_{x \to 0}\dfrac{2\sin^{2}\left(\dfrac{3x}{2}\right)}{\dfrac{4}{9}{\left(\dfrac{3x}{2}\right)}^{2}}\right)$

$= f\left(\dfrac{9}{2}\lim_{x \to 0}\dfrac{\sin^{2}\left(\dfrac{3x}{2}\right)}{{\left(\dfrac{3x}{2}\right)}^{2}}\right)$

$= f\left(\dfrac{9}{2} \times {1}^{2}\right)\left\{∵ \lim_{x \to 0}\dfrac{\sin x}{x} = 1\right\}$

$= f\left(\dfrac{9}{2}\right)$

$= \dfrac{2}{9}$